The Parker Instability under a Linear Gravity

A linear stability analysis has been done to a magnetized disk under a linear gravity. We have reduced the linearized perturbation equations to a second-order differential equation which resembles the Schr"{o}dinger equation with the potential of a harmonic oscillator. Depending on the signs of energy and potential terms, eigensolutions can be classified into continuum'' and discrete’’ families. When magnetic field is ignored, the continuum family is identified as the convective mode, while the discrete family as acoustic-gravity waves. If the effective adiabatic index $\gamma$ is less than unity, the former develops into the convective instability. When a magnetic field is included, the continuum and discrete families further branch into several solutions with different characters. The continuum family is divided into two modes: one is the original Parker mode, which is a slow MHD mode modulated by the gravity, and the other is a stable Alfv'en mode. The Parker modes can be either stable or unstable depending on $\gamma$. When $\gamma$ is smaller than a critical value ${\gamma }_{cr}$, the Parker mode becomes unstable. The discrete family is divided into three modes: a stable fast MHD mode modulated by the gravity, a stable slow MHD mode modulated by the gravity, and an unstable mode which is also attributed to a slow MHD mode. The unstable discrete mode does not always exist. Even though the unstable discrete mode exists, the Parker mode dominates it if the Parker mode is unstable. However, if $\gamma \ge {\gamma }_{cr}$, the discrete mode could be the only unstable one. When $\gamma$ is equal ${\gamma }_{cr}$, the minimum growth time of the unstable discrete mode is $1.3\times {10}^8$ years with a corresponding length scale of 2.4 kpc. It is suggestive that the corrugatory features seen in the Galaxy and external galaxies are related to the unstable discrete mode.